From Euclidean to Hilbert Spaces: Introduction to Functional Analysis and Its Applications

ISBN-13: 9781786306821 / Angielski / Twarda / 2021 / 368 str.

Edoardo Provenzi

Spis treści

Preface xiChapter 1. Inner Product Spaces (Pre-Hilbert) 11.1. Real and complex inner products 11.2. The norm associated with an inner product and normed vector spaces 61.2.1. The parallelogram law and the polarization formula 91.3. Orthogonal and orthonormal families in inner product spaces 111.4. Generalized Pythagorean theorem 111.5. Orthogonality and linear independence 131.6. Orthogonal projection in inner product spaces 151.7. Existence of an orthonormal basis: the Gram-Schmidt process 191.8. Fundamental properties of orthonormal and orthogonal bases 201.9. Summary 28Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing 312.1. The space l²(ZN) and its canonical basis 312.1.1. The orthogonal basis of complex exponentials in l²(ZN) 342.2. The orthonormal Fourier basis of l²(ZN) 382.3. The orthogonal Fourier basis of l²(ZN) 402.4. Fourier coefficients and the discrete Fourier transform 412.4.1. The inverse discrete Fourier transform 442.4.2. Definition of the DFT and the IDFT with the orthonormal Fourier basis 462.4.3. The real (orthonormal) Fourier basis 472.5. Matrix interpretation of the DFT and the IDFT 482.5.1. The fast Fourier transform 512.6. The Fourier transform in signal processing 512.6.1. Synthesis formula for 1D signals: decomposition on the harmonic basis 512.6.2. Signification of Fourier coefficients and spectrums of a 1D signal 532.6.3. The synthesis formula and Fourier coefficients of the unit pulse 542.6.4. High and low frequencies in the synthesis formula 552.6.5. Signal filtering in frequency representation 592.6.6. The multiplication operator and its diagonal matrix representation 602.6.7. The Fourier multiplier operator 602.7. Properties of the DFT 612.7.1. Periodicity of Z and ? 622.7.2. DFT and shift 632.7.3. DFT and conjugation 672.7.4. DFT and convolution 682.8. The DFT and stationary operators 732.8.1. The DFT and the diagonalization of stationary operators 752.8.2. Circulant matrices 772.8.3. Exhaustive characterization of stationary operators 782.8.4. High-pass, low-pass and band-pass filters 822.8.5. Characterizing stationary operators using shift operators 832.8.6. Frequency analysis of first and second derivation operators (discrete case) 842.9. The two-dimensional discrete Fourier transform (2D DFT) 882.9.1. Matrix representation of the 2D DFT: Kronecker product versus iteration of two 1D DFTs 912.9.2. Properties of the 2D DFT 932.9.3. 2D DFT and stationary operators 952.9.4. Gradient and Laplace operators and their action on digital images 972.9.5. Visualization of the amplitude spectrum in 2D 972.9.6. Filtering: an example of digital image filtering in a Fourier space 1002.10. Summary 102Chapter 3. Lebesgue's Measure and Integration Theory 1053.1. Riemann versus Lebesgue 1053.2. sigma-algebra, measurable space, measures and measured spaces 1063.3. Measurable functions and almost-everywhere properties (a.e) 1083.4. Integrable functions and Lebesgue integrals 1093.5. Characterization of the Lebesgue measure on R and sets with a null Lebesgue measure 1113.6. Three theorems for limit operations in integration theory 1133.7. Summary 114Chapter 4. Banach Spaces and Hilbert Spaces 1154.1. Metric topology of inner product spaces 1164.2. Continuity of fundamental operations in inner product spaces 1204.2.1. Equivalence of separated topologies in finite-dimension vector spaces 1284.3. Cauchy sequences and completeness: Banach and Hilbert 1294.3.1. Completeness of vector spaces 1334.3.2. Characterizing the completeness of normed vector spaces using series 1354.3.3. Banach fixed-point theorem 1394.4. Remarkable examples of Banach and Hilbert spaces 1454.4.2. L¯oo and l¯oo spaces 1564.4.3. Inclusion relationships between l¯p spaces 1614.4.4. Inclusion relationships between L¯p spaces 1634.4.5. Density theorems in L¯p(X,A,mu) 1654.5. Summary 169Chapter 5. The Geometric Structure of Hilbert Spaces 1715.1. The orthogonal complement in a Hilbert space and its properties 1715.2. Projection onto closed convex sets: theorem and consequences 1745.2.1. Characterization of closed vector subspaces in Hilbert spaces 1805.3. Polar and bipolar subsets of a Hilbert space 1825.4. The (orthogonal) projection theorem in a Hilbert space 1855.5. Orthonormal systems and Hilbert bases 1885.5.1. Bessel's inequality and Fourier coefficients 1895.5.2. The Fischer-Riesz theorem 1925.5.3. Characterizations of a Hilbert basis (or complete orthonormal system) 1945.5.4. Isomorphisms between Hilbert spaces 1995.5.5. l²(N,K) as the prototype of separable Hilbert spaces of infinite dimension 2015.6. The Fourier Hilbert basis in L¯2 2025.6.1. L²[-pi, pi] or L²[0, 2pi] 2025.6.2. L²(T) 2045.6.3. L²[a, b] 2055.6.4. Real Fourier series 2065.6.5. Pointwise convergence of the real Fourier series: Dirichlet's theorem 2125.6.6. The Gibbs phenomenon and Cesàro summation 2145.6.7. Speed of convergence to 0 of Fourier coefficients 2145.6.8. Fourier transform in L²(T) and shift 2185.7. Summary 219Chapter 6. Bounded Linear Operators in Hilbert Spaces 2216.1. Fundamental properties of bounded linear operators between normed vector spaces 2236.1.1. Continuity of linear operators defined on a finite-dimensional normed vector space 2266.2. The operator norm, convergence of operator sequences and Banach algebras 2276.2.1. A classical example of a non-bounded linear operator on a vector space of infinite dimension 2386.3. Invertibility of linear operators 2396.4. The dual of a Hilbert space and the Riesz representation theorem 2446.4.1. The scalar product induced on the dual of a Hilbert space 2496.5. Bilinear forms, sesquilinear forms and associated quadratic forms 2496.5.1. The Lax-Milgram theorem and its consequences 2576.6. The adjoint operator: presentation and properties 2616.7. Orthogonal projection operators in a Hilbert space 2696.7.1. Bounded multiplication operators and their relation to orthogonal projectors 2786.7.2. Geometric realization of orthogonal projection operators via orthonormal systems 2806.8. Isometric and unitary operators 2866.8.1. Characterizations of isometric and unitary operators 2886.8.2. Relationship between isometric and unitary operators and orthonormal systems 2936.9. The Fourier transform on S(R¯n), L¹(R¯n) and L²(R¯n) 2966.9.1. The invariance of the Schwartz space with respect to the Fourier transform 2966.9.2. Extension of the Fourier transform of S(R¯n) to L¹(R¯n): the Riemann-Lebesgue theorem 3016.9.3. Extension of the Fourier transform to a unitary operator on L²(R¯n): the Fourier-Plancherel transform 3026.9.4. Relationship between the Fourier-Plancherel transform and the Hermitian Hilbert basis 3056.9.5. The Fourier transform and convolution 3066.9.6. Convolution and Fourier transforms in L²: localization of the Fourier transform 3096.10. The Nyquist-Shannon sampling theorem 3106.10.1. The Nyquist frequency: aliasing and oversampling 3126.11. Application of the Fourier transform to solve ordinary and partial differential equations 3136.11.1. Solving an ordinary differential equation using the Fourier transform 3136.11.2. The Fourier transform and partial differential equations 3156.11.3. Solving the partial differential equation for heat propagation using the Fourier transform 3166.12. Summary 319Appendix 1: Quotient Space 323Appendix 2: The Transpose (or Dual) of a Linear Operator 329Appendix 3: Uniform, Strong and Weak Convergence 331References 335Index 337

O autorze

Edoardo Provenzi is Professor of Mathematics at the University of Bordeaux, France. He studies visual phenomena and their applications in image processing and computer vision, employing tools from differential geometry, harmonic analysis and mathematical physics.